Solving an integral using the error function

Question

I want to evaluate the following integral $$ \int{\frac{1}{\sqrt{2\pi t}} \exp(\frac{-1}{2t}((a-x-wt)^2))dt}$$ where $w>0, 0<x<a$. Using WolframAlpha I obtain an expression for this indefinite integral including terms with the error function $$ \frac{1}{2w} \left(\text{erf}\left(\frac{tw-(a-x)}{\sqrt{2t}}\right) +\exp(2w(a-x)) \text{erf}\left(\frac{tw+(a-x)}{\sqrt{2t}}\right) - \exp(2w(a-x))+1\right)$$ I’m interested in a step-by-step solution. Especially, I don’t know where to substitute the error function exactly. Thanks!